Using Lemma 7.2 [1]}, we have
\(\left(\mathbb {E}\sup _{s \in [0,t]} \left| \int _0^s (u(\sigma ), d W(\sigma )) \right|\right)^2 \le \mathbb {E}\left(\sup _{s \in [0,t]} \left|\int _0^s (u(\sigma ), d W(\sigma ))\right|\right)^2 = \mathbb {E}\sup _{s \in [0,t]} \left|\int _0^s (u(\sigma ), d W(\sigma )) \right|^2 \le \) 
\(\le C \int _{0}^t \mathbb {E}\Vert u(s)\Vert ^2_H ds \le C \int _{0}^t \mathbb {E}\sup _{\sigma \in [0,s]}(\Vert u(\sigma )\Vert ^2_H + \Vert w(\sigma )\Vert ^2_H) ds.\) 
